Complete reduplication is used with a handful of English words for intensive effect: very very good; an old old man; big big trouble.
It seems natural to use it for intensification or multiplicity. In Indonesian it can be used freely on nouns to indicate a diverse plural. Plurality is not normally marked (ada orang disini = 'there is a person here' or 'there are people here'), so ada orang-orang disini means 'there are all sorts of people here'. Doubled words such as orang-orang are sometimes written as orang2.
Partly reduplicative compounds are also widespread. In English, mumbo-jumbo and flip-flop have already been mentioned; we might also mention itsy-bitsy and eency-weency. What seems to be common to them is that they're never entirely serious words. We also have quite a few alliterative expressions like kith and kin, home and hosed, which are not reduplications as such but clearly home in on the same instinct for repetition. So do other languages: Turkish is especially fond of almost-reduplicative idioms, e.g. çoluk çocuk 'wife and kids', boy bos 'size and shape'. In Turkish also, sound symbolic words often come reduplicated: horul horul horlamak 'to snore like a pig'.
The original use of 'reduplication' as a grammatical term was in the addition of syllables in the classical languages to indicate the perfect tense. In Greek, almost all verbs form their perfect with a reduplication of the initial consonant, followed by the vowel e. There may be internal changes as well, and the endings for the perfect are different from those for the present: luô 'I loose', leluka 'I loosed'. Many sound changes peculiar to Greek have added layers of complication to this: one is that you don't get two aspirated consonants together (Grassmann's Law), so phileô 'I love' becomes not *phephilêka but pephilêka. Other changes result from the disappearance of consonants such as w and s.
In Latin it only applied to a certain number of common verbs: tango 'I touch', tetigi 'I touched', compared to more regular video 'I see', vidi 'I saw'. Changes in the Latin root include the fact that the present tense sometimes has an infix -n (tango not tago) and the vowel in the perfect is changed (-tag- to -tig-) by the fact that in Old Latin there was a strong initial stress. The reduplicated vowel is usually -e- but may be -o-.
In Sanskrit likewise: tan- 'stretch' becomes tatan- in the perfect. The fact that the -a- in the reduplicated syllable comes from an earlier -e- is shown by the behaviour of stems beginning with velar sounds: kar- 'make' becomes cakar-, and gam- 'go' becomes jagam-. The change of g- to j- took place before the front vowel -e-, which subsequently changed to -a-.
As well as the perfect, some verbs have reduplicated forms with a vowel -i in the present tense, as in Greek gignosko 'I know' (cf. gnosis 'knowledge'). These two Indo-European reduplications also occurred in the old Germanic language Gothic, but have not survived into modern Germanic: there are no examples in English.
Syllabic reduplication is not restricted to the Indo-European languages. It is common in Austronesian languages: for example in Samoan, adjectives generally form their plural by reduplicating the stressed syllable: tele 'great', tetele; or lapo'a 'big', lapopo'a. In the Afro-Asiatic language Somali it marks plurality on some classes of adjective and noun: whole syllable in adj. cusub 'new', pl. cuscusub, and consonant only in noun miis 'table', pl. miisas.